★Approximation Functions

f(x) = (ax + bP) / (bx + a) … Basic Formula
where a, b, and P are positive integers. a/b > √P

Properties of the Basic Formula
1. √P = f(√P) … Has a fixed point at √P.
2. f(∞)・f(0)＝P … When x is a positive integer,
　　the rational function f(x) yields an approximate rational value of the fixed point within the range f(∞) > f(x) > f(0).
3. Let F₁(x) = f(x) and F_n(x) = F_(n-1)(f(x)). Then, F_n(x) = (Ax + BP) / (Bx + A). F_n(x) also possesses the same properties as f(x).
4. F_n(∞) − F_n(0) < F_{n-1}(∞) − F_{n-1}(0)
　　... As the number of substitutions of x → f(x) increases, the range of approximation to the fixed point narrows.
5. When b = 1, the basic formula can be rewritten as f(x) = a − (a² − P)/(a + x), and Fn(x) yields an approximate continued fraction.

Using the properties described above, we derive more complex expressions by repeatedly applying substitutions starting from a simple basic form.
Starting with a = (the integer part of √P) + 1 and b = 1, we compute F_n(x) = (Ax + BP) / (Bx + A).